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# primitive element theorem

###### Theorem 1.

Let $F$ and $K$ be arbitrary fields, and let $K$ be an extension of $F$ of finite degree. Then there exists an element $\alpha\in K$ such that $K=F(\alpha)$ if and only if there are finitely many fields $L$ with $F\subseteq L\subseteq K$.

Note that this implies that every finite separable extension is not only finitely generated, it is generated by a single element.

Let $X$ be an indeterminate. Then $\mathbb{Q}(X,i)$ is not generated over $\mathbb{Q}$ by a single element (and there are infinitely many intermediate fields $\mathbb{Q}(X,i)/L/\mathbb{Q}$). To see this, suppose it is generated by an element $\alpha$. Then clearly $\alpha$ must be transcendental, or it would generate an extension of finite degree. But if $\alpha$ is transcendental, we know it is isomorphic to $\mathbb{Q}(X)$, and this field is not isomorphic to $\mathbb{Q}(X,i)$: for example, the polynomial $Y^{2}+1$ has no roots in the first but it has two roots in the second. It is also clear that it is not sufficient for every element of $K$ to be algebraic over $F$: we know that the algebraic closure of $\mathbb{Q}$ has infinite degree over $\mathbb{Q}$, but if $\alpha$ is algebraic over $\mathbb{Q}$ then $[\mathbb{Q}(\alpha):\mathbb{Q}]$ will be finite.

This theorem has the corollary:

###### Corollary 1.

Let $F$ be a field, and let $[F(\beta,\gamma):F]$ be finite and separable. Then there exists $\alpha\in F(\beta,\gamma)$ such that $F(\beta,\gamma)=F(\alpha)$. In fact, we can always take $\alpha$ to be an $F$-linear combination of $\beta$ and $\gamma$.

To see this (in the case of characteristic $0$), we need only show that there are finitely many intermediate fields. But any intermediate field is contained in the splitting field of the minimal polynomials of $\beta$ and $\gamma$, which is Galois with finite Galois group. The explicit form of $\alpha$ comes from the proof of the theorem.

For more detail on this theorem and its proof see, for example, *Field and Galois Theory*, by Patrick Morandi (Springer Graduate Texts in Mathematics 167, 1996).

## Mathematics Subject Classification

12F05*no label found*65-01

*no label found*

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